Friday, 24 August 2012

Multiplier theory: one is the magic number

I have written a bit about multipliers, particularly of the
balanced budget kind, but judging by comments some recap and elaboration may be
useful. So here is why, for all government spending multipliers, one is the
number to start from. To make it a bit of a challenge (for me), I’ll not use any
algebra.

Any discussion has to be context specific. Imagine a two
period world. The first period is demand deficient because interest rates are
stuck at the zero lower bound[1],
but in the (longer) second period monetary policy ensures output is fixed at
some level independent of aggregate demand (i.e. its supply determined). Government
spending increases in period 1 only. That is the context when these multipliers
are likely to be important as a policy tool.

1) Balanced budget multiplier

To recap,
for a balanced budget multiplier (BBM), here is a simple proof in terms of
sector balances for a closed economy. A BBM by definition does not change the
public sector’s finance balance (FB). It seems very reasonable to assume that consumers
consume a proportion less than one of any change to their first period post-tax
income. So if higher taxes reduced income, their consumption falls by less, so
their FB moves into deficit. But as the sum of the public and private sector’s
FB sums to zero, it cannot do this. So post-tax income cannot fall. Hence
pre-tax income must rise to just offset the impact of higher taxes. The BBM is
one.

The nice thing about this result is that it holds whatever
fraction of current income is consumed (as long as it’s less than one), so it
is independent of the degree of consumption smoothing. What about lower
consumption in the second period? No need to worry, as monetary policy ensures
demand is adequate in the second period.

Although a good place to start, allowing for an impact on
expected inflation and therefore real interest rates will raise this number
above one. In addition, as DeLong
and Summers discuss, hysteresis effects will also raise period 2 output and
income from the supply side, some of which consumers will consume in period 1.
We would get similar effects if the higher government spending was in the form
of useful intrastructure investment. So in this case one is the place to start,
but it looks like a lower bound.

2) BBM in an open economy

I’m still seeing people claim that the BBM in an open
economy is small. It could be, if the government acts foolishly. Suppose the
government increases its spending entirely on defence, which in turn consists
of buying a new fighter jet from an overseas country. The impact on the demand
for domestic output is zero. But consumers are paying for this through higher
taxes, so their spending decreases – we get a negative multiplier.

Now consider the opposite: the additional government
spending involves no imported goods whatsoever. The multiplier is one. You can
do the maths, but it is easy to show that this is a solution by thinking about
the BBM in a closed economy. There consumption does not change, because a BBM=1
raises pre-tax income to offset higher taxes. But if consumption does not
change, neither will imports, so this is also the solution in the open economy
case.

What the textbooks do is apply a marginal propensity to
import to total output, which implicitly assumes that the same proportion of
government spending is imported as consumption spending. For most economies
that is not the case, as the ‘home bias’ for government spending is much
larger. Furthermore, if the government is increasing its spending with the aim
of raising output, it can choose to spend it on domestically produced output
rather than imports. So, a multiplier of one is again a good place to start.
Allowing some import leakage will reduce the multiplier, but this could easily
be offset by the real interest rate effects discussed above, particular as
these would in an open economy depreciate the real exchange rate.

3) Debt financed government spending with future tax
increases

Although this is the standard case, from a pedagogical point
of view I think it’s better to start with the BBM, and note that it’s all the
same with Ricardian Equivalence. We can then have a discussion about which are
the quantitatively important reasons why Ricardian Equivalence does not hold.
All these go to raise the multiplier above one. You have to add, however, some
discussion about the impact that distortionary tax increases will have on
output in the second period, which reduces second period output and, through consumption
smoothing, the size of the first period multiplier.

4) Debt financed government spending without tax increases

In an earlier post
I queried why arguments for the expansionary impact of government spending
increases always involved raising taxes at some point. For debt finance, why
not assume lower government spending in the future rather than higher taxes.
The advantage is that you do not need to worry about supply side tax effects. Monetary
policy ensures there is no impact on output of lower government spending in the
second period. Now, unlike the BBM case, we do need to make some assumptions
about the degree of consumption smoothing. If you think the first period is
short enough, and consumers smooth enough, such that the impact of higher
income on consumption in the first period is negligible, then we have a
multiplier of one again.

[1]I
assume Quantitative Easing cannot negate the ZLB problem, and that inflation
targets are in place and fixed. This is not about fiscal stimulus versus NGDP
targeting, but just about macro theory.

5 comments:

The result rather depends on the limitation to two periods (or any fixed number), I think. Since this excludes the possibility that transaction rates (and money velocity) are increased - perhaps due to a confidence effect.

But this also goes to show that numbers can only tell you so much. What you spend on is also important, not just how much, in determining the paths of future revenue, expenditure and welfare.

But maybe we cannot expect governments to get these 'qualitative' spending decisions right? Which would support your point that one should be the assumption.

I am confused about whether you are talking of real or nominal income. If there is no output gap, then a balanced budget multiplier cannot be 1 in real terms, it can only be one in nominal terms. Then you have a fiscal theory of the price level, right?

Or what am I getting wrong? Why doesn't your argument work just as well if we are at full output in period 1?

Mmm. Also, if (5) is government debt being inflated away, make (6) repudiation.

Analysis of repudiation is hard, but repudiation is common and an important case to analyze; it seems to work out really well, for the most part. This probably has to do with *who tends to own government bonds*; repudiation acts as a wealth transfer from the rich to the poor, and that usually boosts the economy due to marginal propensity to spend.

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